Open Access
November 2019 Limiting saddlepoint relative errors in large deviation regions under purely Tauberian conditions
Ronald W. Butler, Andrew T.A. Wood
Bernoulli 25(4B): 3379-3399 (November 2019). DOI: 10.3150/18-BEJ1093


Most theoretical results on the relative errors of saddlepoint approximations in the extreme tails have involved placing conditions on the density/mass function. Checking the validity of such conditions is problematic when density/mass functions are intractable, as is typically the case in important practical applications involving convolved, compound, and first-passage distributions as well as for moment generating functions MGFs that are regularly varying. In this paper, we present novel conditions which ensure the existence of positive finite limiting relative errors for saddlepoint density/mass function and survival function approximations. These conditions, which are rather weak, are expressed entirely in terms of the MGF, hence the description purely Tauberian. We focus mainly on the cases in which there are positive and negative gamma distributional limits (the only other non-degenerate possibility being a Gaussian limit) and we show how to check the new conditions in important classes of models in these two settings.


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Ronald W. Butler. Andrew T.A. Wood. "Limiting saddlepoint relative errors in large deviation regions under purely Tauberian conditions." Bernoulli 25 (4B) 3379 - 3399, November 2019.


Received: 1 December 2017; Revised: 1 July 2018; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110141
MathSciNet: MR4010958
Digital Object Identifier: 10.3150/18-BEJ1093

Keywords: compound distribution , first-passage distribution , regular variation , saddlepoint approximation , Tauberian arguments

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4B • November 2019
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